In numerical computations of response properties of electronic systems, the standard model is Kohn–Sham density functional theory (KS-DFT). Here we investigate the mathematical status of the simplest class of excitations in KS-DFT, HOMO–LUMO excitations. We show that such excitations, i.e. excited states of the Kohn–Sham Hamiltonian, exist for $Z>N$, where $Z$ is the total nuclear charge and $N$ is the number of electrons. The result applies under realistic assumptions on the exchange–correlation functional, which we verify explicitly for the widely used PZ81 and PW92 functionals. By contrast, and somewhat surprisingly, we find using a method of Glaser, Martin, Grosse, and Thirring (Glaser et al., 1976) that in case of the hydrogen and helium atoms, excited states do not exist in the neutral case $Z=N$ when the self-consistent KS ground state density is replaced by a realistic but easier to analyze approximation (in case of hydrogen, the true Schrödinger ground state density). Implications for interpreting minus the HOMO eigenvalue as an approximation to the ionization potential are indicated.

在电子系统响应特性的数值计算中，标准模型为Kohn-Sham密度泛函理论（KS-DFT）。在这里，我们研究了KS-DFT，HOMO-LUMO激发中最简单的激发类型的数学状态。我们证明了这样的激发，即Kohn-Sham哈密顿量的激发态存在于$ž>ñ$，在哪里 $ž$ 是总核电荷， $ñ$是电子数。该结果适用于交换相关函数的实际假设，我们对广泛使用的PZ81和PW92函数进行了明确验证。相比之下，有些令人惊讶的是，我们发现使用Glaser，Martin，Grosse和Thirring（Glaser等人，1976年）的方法发现，在氢原子和氦原子的情况下，中性情况下不存在激发态$ž=ñ$当将自洽的KS基态密度替换为现实但更易于分析的近似值时（对于氢，则为真正的薛定ding基态密度）。指出了将负HOMO特征值解释为电离势的近似值的含义。